A pharmacist has two vitamin-supplement powders. The first powder is 20% vitamin B1 and 30% vitamin B2. The second is 15% vitamin B1 and 10% vitamin B2. How many milligrams of each powder should the pharmacist use to make a mixture that contains 140 mg of vitamin B1 and 190 mg of vitamin B2?

first powder

mg

second powder

mg

Let's denote:

- \( x \) = milligrams of the first powder.
- \( y \) = milligrams of the second powder.

We know from the problem that:

- The first powder contains 20% vitamin B1, meaning \( 0.20x \) milligrams of vitamin B1 come from the first powder.
- The second powder contains 15% vitamin B1, meaning \( 0.15y \) milligrams of vitamin B1 come from the second powder.

Similarly:

- The first powder contains 30% vitamin B2, meaning \( 0.30x \) milligrams of vitamin B2 come from the first powder.
- The second powder contains 10% vitamin B2, meaning \( 0.10y \) milligrams of vitamin B2 come from the second powder.

The total amount of vitamin B1 in the mixture is 140 mg, and the total amount of vitamin B2 in the mixture is 190 mg.

We can now set up the system of equations:

\[
\begin{aligned}
0.20x + 0.15y &= 140 \quad \text{(for vitamin B1)} \\
0.30x + 0.10y &= 190 \quad \text{(for vitamin B2)} 
\end{aligned}
\]

### Step 1: Solve for one variable

First, let's multiply the second equation by 3 to eliminate decimals:

\[
0.90x + 0.30y = 570
\]

Now subtract the original second equation \( 0.30x + 0.10y = 190 \) from this new equation:

\[
(0.90x + 0.30y) - (0.30x + 0.10y) = 570 - 190
\]

This simplifies to:

\[
0.60x + 0.20y = 380
\]

### Step 2: Solve for \( x \)

Now let's solve the first equation for \( y \) by isolating \( y \) in terms of \( x \):

\[
0.20x + 0.15y = 140
\]

Multiply the entire equation by 10:

\[
2x + 1.5y = 1400
\]We can proceed by solving the system step by step. Let's recap:

We already derived the two equations:

1. \( 0.20x + 0.15y = 140 \)
2. \( 0.30x + 0.10y = 190 \)

To avoid decimals, we can multiply the first equation by 100 and the second by 100 as well:

\[
20x + 15y = 14000
\]
\[
30x + 10y = 19000
\]

Now, we solve these equations simultaneously. Start by eliminating one variable. Let's multiply the second equation by 1.5 to match the coefficients of \( y \):

\[
45x + 15y = 28500
\]

Now, subtract the first equation from this:

\[
(45x + 15y) - (20x + 15y) = 28500 - 14000
\]

Simplifying this gives:

\[
25x = 14500
\]

\[
x = \frac{14500}{25} = 580
\]

Now that we have \( x = 580 \), substitute this back into one of the original equations, say:

\[
20(580) + 15y = 14000
\]

This simplifies to:

\[
11600 + 15y = 14000
\]

\[
15y = 14000 - 11600 = 2400
\]

\[
y = \frac{2400}{15} = 160
\]

### Final Answer:
- The pharmacist should use **580 mg** of the first powder.
- The pharmacist should use **160 mg** of the second powder.

Would you like further details or have any questions?

---

Here are 5 related questions to explore:
1. How would the result change if the total amount of vitamin B1 was 150 mg instead of 140 mg?
2. What if the vitamin B2 percentage in the second powder was increased to 15%?
3. Can you set up a similar problem where both powders contain three vitamins instead of two?
4. How would you solve this system using matrix methods?
5. What happens if you change the total quantity of vitamin B1 and B2 proportionally?

**Tip:** When solving systems of equations with decimals, multiplying by powers of 10 helps to eliminate fractional coefficients.

A pharmacist has two vitamin-supplement powders. The first powder is 20% vitamin B1 and 30% vitamin B2. The second is 15% vitamin B1 and 10% vitamin B2. How many milligrams of each powder should the pharmacist use to make a mixture that contains 140 mg of vitamin B1 and 190 mg of vitamin B2?

This detailed solution explains how to solve a real-world problem involving two vitamin-supplement powders. The problem requires using algebraic techniques, including linear systems of equations, to find how many milligrams of each powder yield 140 mg of vitamin B1 and 190 mg of vitamin B2. Suitable for high school students in Grades 10-12.

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